Estimation of a normal mean relative to balanced loss functions

Abstract

LetX ₁,…,X _nbe a random sample from a normal distribution with mean θ and variance σ². The problem is to estimate θ with Zellner's (1994) balanced loss function,$L_B \left( {\hat \theta ,\theta } \right) = \frac{\omega }{n}\sum {_1^n \left( {X_i - \hat \theta } \right)^2 } + \left( {1 - w} \right)\left( {\theta ,\hat \theta } \right)^2$% MathType!End!2!1!, where 0<ω<1. It is shown that the sample mean$\bar X$% MathType!End!2!1!, is admissible. More generally, we investigate the admissibility of estimators of the form$\alpha \bar X + b$% MathType!End!2!1! under$L_B \left( {\hat \theta ,\theta } \right)$% MathType!End!2!1!. We also consider the weighted balanced loss function,$L_W \left( {\hat \theta ,\theta } \right) = \omega q\left( \theta \right)\frac{{\sum {_1^n \left( {X_i - \hat \theta } \right)^2 } }}{n} + \left( {1 - w} \right)q\left( \theta \right)\left( {\theta ,\hat \theta } \right)^2$% MathType!End!2!1!, whereq(θ) is any positive function of θ, and the class of admissible linear estimators is obtained under such loss withq(θ) =e ^θ.